with \(\rho \in (-1,1)\setminus \lbrace 0\rbrace \)  is called the (exponentially decreasing for \(x\rightarrow \infty \) ) Bloch mode of \(L\)  and \(\rho \)  is called the Floquet multiplier. The existence of \(\Phi \)  is guaranteed by the assumption that \(0\notin \sigma (L)\) .
This is essentially Hill's theorem, cf. [1]}. Note that \(\Psi (x)\Phi (-x)\)  is a second Bloch mode of \(L\) , which is exponentially increasing for \(x\rightarrow \infty \) . The functions \(\Phi \)  and \(\Psi \)  form a fundamental system of solutions for operator \(L\)  on \(\mathbb {R}\) . Next we explain how \(\Phi \)  is constructed, why it can be taken real-valued and why it does not vanish at \(x=0\)  so that we can assume w.l.o.g \(\Phi (0)=1\) .
